117 chap7 slides (1)

Chapter 7

Hypothesis Testing with One Sample

1Math 117 – Eddie Laanaoui

Chapter Outline

• 7.1 Introduction to Hypothesis Testing

• 7.2 Hypothesis Testing for the Mean (Large Samples)

• 7.3 Hypothesis Testing for the Mean (Small Samples)

• 7.4 Hypothesis Testing for Proportions


Section 7.1

Introduction to Hypothesis Testing


Section 7.1 Objectives

• State a null hypothesis and an alternative hypothesis

• Identify type I and type II errors and interpret the level of significance

• Determine whether to use a one-tailed or two-tailed statistical test and find a p-value

• Make and interpret a decision based on the results of a statistical test

• Write a claim for a hypothesis test


Hypothesis Tests

Hypothesis test

• A process that uses sample statistics to test a claim about the value of a population parameter.

• For example: An automobile manufacturer advertises that its new hybrid car has a mean mileage of 50 miles per gallon. To test this claim, a sample would be taken. If the sample mean differs enough from the advertised mean, you can decide the advertisement is wrong.


Hypothesis Tests

Statistical hypothesis

• A statement, or claim, about a population parameter.

• Need a pair of hypotheses

• one that represents the claim

• the other, its complement

• When one of these hypotheses is false, the other must be true.

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Math 117 – Eddie Laanaoui

Stating a Hypothesis

Null hypothesis

• A statistical hypothesis that contains a statement of equality such as ≤, =, or ≥.

• Denoted H0 read “H subzero” or “H naught.”

Alternative hypothesis

• A statement of inequality such as >, ≠, or <.

• Must be true if H0 is false.

• Denoted Ha read “H sub-a.”

complementary statements


Stating a Hypothesis

• To write the null and alternative hypotheses, translate the claim made about the population parameter from a verbal statement to a mathematical statement.

• Then write its complement.

H0: μ ≤ kHa: μ > k

H0: μ ≥ kHa: μ < k

H0: μ = kHa: μ ≠ k

• Regardless of which pair of hypotheses you use, you always assume μ = k and examine the sampling distribution on the basis of this assumption.


Example: Stating the Null and Alternative Hypotheses

Write the claim as a mathematical sentence. State the null and alternative hypotheses and identify which represents the claim.1.A university publicizes that the proportion of its students who graduate in 4 years is 82%.

Equality condition

Complement of H0

H0:

Ha:

(Claim)p = 0.82

p ≠ 0.82

Solution:


μ ≥ 2.5 gallons per minute


Write the claim as a mathematical sentence. State the null and alternative hypotheses and identify which represents the claim.2.A water faucet manufacturer announces that the mean flow rate of a certain type of faucet is less than 2.5 gallons per minute.

Inequality condition

Complement of HaH0:

Ha:(Claim)μ < 2.5 gallons per minute

Solution:


μ ≤ 20 ounces


Write the claim as a mathematical sentence. State the null and alternative hypotheses and identify which represents the claim.3.A cereal company advertises that the mean weight of the contents of its 20-ounce size cereal boxes is more than 20 ounces.

Inequality condition

Complement of HaH0:

Ha:(Claim)μ > 20 ounces

Solution:


Types of Errors

• No matter which hypothesis represents the claim, always begin the hypothesis test assuming that the equality condition in the null hypothesis is true.

• At the end of the test, one of two decisions will be made:

reject the null hypothesis

fail to reject the null hypothesis

• Because your decision is based on a sample, there is the possibility of making the wrong decision.


Types of Errors

• A type I error occurs if the null hypothesis is rejected when it is true.

• A type II error occurs if the null hypothesis is not rejected when it is false.

Actual Truth of H0

Decision H0 is true H0 is false

Do not reject H0 Correct Decision Type II Error

Reject H0 Type I Error Correct Decision


Example: Identifying Type I and Type II Errors

The USDA limit for salmonella contamination for chicken is 20%. A meat inspector reports that the chicken produced by a company exceeds the USDA limit. You perform a hypothesis test to determine whether the meat inspector’s claim is true. When will a type I or type II error occur? Which is more serious? (Source: United States Department of Agriculture)


Let p represent the proportion of chicken that is contaminated.

Solution: Identifying Type I and Type II Errors

H0:

Ha:

p ≤ 0.2

p > 0.2

Hypotheses:

(Claim)

0.16 0.18 0.20 0.22 0.24p

H0: p ≤ 0.20 H1: p > 0.20

Chicken meets USDA limits.

Chicken exceeds USDA limits.



A type I error is rejecting H0 when it is true.

The actual proportion of contaminated chicken is lessthan or equal to 0.2, but you decide to reject H0.

A type II error is failing to reject H0 when it is false.

The actual proportion of contaminated chicken is greater than 0.2, but you do not reject H0.

H0:Ha:

p ≤ 0.2p > 0.2

Hypotheses:(Claim)



H0:Ha:

p ≤ 0.2p > 0.2

Hypotheses:(Claim)

• With a type I error, you might create a health scare and hurt the sales of chicken producers who were actually meeting the USDA limits.

• With a type II error, you could be allowing chicken that exceeded the USDA contamination limit to be sold to consumers.

• A type II error could result in sickness or even death.


Level of Significance

Level of significance: α • Your maximum allowable probability of making a

type I error. P(type I error) = α (alpha).

• By setting the level of significance at a small value, you are saying that you want the probability of rejecting a true null hypothesis to be small.

• Commonly used levels of significance: α = 0.10 α = 0.05 α = 0.01

• P(type II error) = β (beta)


Statistical Tests

• After stating the null and alternative hypotheses and specifying the level of significance, a random sample is taken from the population and sample statistics are calculated.

• The statistic that is compared with the parameter in the null hypothesis is called the test statistic.

x

z (Section 7.4)pt (Section 7.3 n < 30)z (Section 7.2 n ≥ 30)μ

Standardized test statistic

Test statisticPopulation parameter

p̂


P-values

P-value (or probability value)

• The probability, if the null hypothesis is true, of obtaining a sample statistic with a value as extreme or more extreme than the one determined from the sample data.

• Depends on the nature of the test.


Nature of the Test

• Three types of hypothesis tests left-tailed test right-tailed test two-tailed test


Left-tailed Test

• The alternative hypothesis Ha contains the less-than inequality symbol (<).

z0 1 2 3-3 -2 -1

Test statistic

H0: μ ≥ k

Ha: μ < k

P is the area to the left of the test statistic.


• The alternative hypothesis Ha contains the greater-than inequality symbol (>).

z0 1 2 3-3 -2 -1

Right-tailed Test

H0: μ ≤ k

Ha: μ > k

Test statistic

P is the area to the right of the test statistic.


Two-tailed Test

• The alternative hypothesis Ha contains the not equal inequality symbol (≠). Each tail has an area of ½P.

z0 1 2 3-3 -2 -1

Test statistic

Test statistic

H0: μ = k

Ha: μ ≠ kP is twice the area to the left of the negative test statistic.

P is twice the area to the right of the positive test statistic.


Example: Identifying The Nature of a Test

For each claim, state H0 and Ha. Then determine whether the hypothesis test is a left-tailed, right-tailed, or two-tailed test. Sketch a normal sampling distribution and shade the area for the P-value.1.A university publicizes that the proportion of its students who graduate in 4 years is 82%.

H0:Ha:

p = 0.82p ≠ 0.82

Two-tailed testz

0-z z

½ P-value½ P-value

Solution:



For each claim, state H0 and Ha. Then determine whether the hypothesis test is a left-tailed, right-tailed, or two-tailed test. Sketch a normal sampling distribution and shade the area for the P-value.2.A water faucet manufacturer announces that the mean flow rate of a certain type of faucet is less than 2.5 gallons per minute.

H0:Ha:

Left-tailed testz

0-z

P-valueμ ≥ 2.5 gpmμ < 2.5 gpm

Solution:



For each claim, state H0 and Ha. Then determine whether the hypothesis test is a left-tailed, right-tailed, or two-tailed test. Sketch a normal sampling distribution and shade the area for the P-value.3.A cereal company advertises that the mean weight of the contents of its 20-ounce size cereal boxes is more than 20 ounces.

H0:Ha:

Right-tailed testz

0 z

P-valueμ ≤ 20 ozμ > 20 oz

Solution:


Making a Decision

Decision Rule Based on P-value• Compare the P-value with α.

If P ≤ α, then reject H0.

If P > α, then fail to reject H0.

Claim

Decision Claim is H0 Claim is Ha

Fail to reject H0

Reject H0

There is not enough evidence to reject the claim

There is enough evidence to reject the claim

There is not enough evidence to support the claim

There is enough evidence to support the claim


z0

Steps for Hypothesis Testing

1. State the claim mathematically and verbally. Identify the null and alternative hypotheses.

H0: ? Ha: ?2. Specify the level of significance.

α = ?3. Calculate the test statistic

and its standardized value, Z.

z0

Test statistic

This sampling distribution is based on the assumption that H0 is true.


x

Steps for Hypothesis Testing

5. Find the P-value.

6. Use the following decision rule.

7. Write a statement to interpret the decision in the context of the original claim.

Is the P-value less than or equal to the level of significance?

Fail to reject H0.

Yes

Reject H0.

No


Section 7.1 Summary

• Stated a null hypothesis and an alternative hypothesis

• Identified type I and type I errors and interpreted the level of significance

• Determined whether to use a one-tailed or two-tailed statistical test and found a p-value

• Made and interpreted a decision based on the results of a statistical test

• Wrote a claim for a hypothesis test


Section 7.2

Hypothesis Testing for the Mean (Large Samples)



• Find P-values and use them to test a mean μ

• Use P-values for a z-test


Using P-values to Make a Decision

Decision Rule Based on P-value

• To use a P-value to make a conclusion in a hypothesis test, compare the P-value with α.

1. If P ≤ α, then reject H0.

2. If P > α, then fail to reject H0.


Example: Interpreting a P-value

The P-value for a hypothesis test is P = 0.0237. What is your decision if the level of significance is

1.0.05?

2.0.01?

Solution:Because 0.0237 < 0.05, you should reject the null hypothesis.

Solution:Because 0.0237 > 0.01, you should fail to reject the null hypothesis.


Finding the P-value

After determining the hypothesis’ standardized test statistic and the test statistic’s corresponding area, do one of the following to find the P-value.

a.For a left-tailed test, P = (Area in left tail).

b.For a right-tailed test, P = (Area in right tail).

c.For a two-tailed test, P = 2(Area in tail of test statistic).


Z-Test for a Mean μ

• Can be used when the population is normal and σ is known, or for any population when the sample size n is at least 30.

• The test statistic is the sample mean

• The standardized test statistic is z

• When n ≥ 30, the sample standard deviation s can be substituted for σ.

xznµ

σ−=

x


Using P-values for a z-Test for Mean μ

1. State the claim mathematically and verbally. Identify the null and alternative hypotheses.

2. Specify the level of significance.

3. Determine the standardized test statistic.

4. Find the area that corresponds to z. (that is the P-value)

State H0 and Ha.

Identify α.

Use NormCdf function in your calculator.

xznµ

σ−=

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In Words In Symbols


Using P-values for a z-Test for Mean μ

Reject H0 if P-value is less than or equal to α. Otherwise, fail to reject H0.

5. Find the P-value.a. For a left-tailed test, P = NormCdf (-E99, z)b. For a right-tailed test,P = NormCdf (z, E99)c. For a two-tailed test, P =2 NormCdf (absolute value of z , E99)

6. Make a decision to reject or fail to reject the null hypothesis.

7. Interpret the decision in the context of the original claim.

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In Words In Symbols


Example: Hypothesis Testing Using P-values

In an advertisement, a pizza shop claims that its mean delivery time is less than 30 minutes. A random selection of 36 delivery times has a sample mean of 28.5 minutes and a standard deviation of 3.5 minutes. Is there enough evidence to support the claim at α = 0.01? Use a P-value.


Solution: Hypothesis Testing Using P-values

• H0:

• Ha:

• α = • Test Statistic:

μ ≥ 30 min

μ < 30 min

0.01

28.5 30

3.5 36

2.57

µ−=

−=

= −

xz

s n• Decision:

At the 1% level of significance, you have sufficient evidence to conclude the mean delivery time is less than 30 minutes.

z0-2.57

0.0051

• P-value = NormCdf(-E99, -2.57)

0.0051 < 0.01 Reject H0


Example: Hypothesis Testing Using P-values

You think that the average franchise investment information shown in the graph is incorrect, so you randomly select 30 franchises and determine the necessary investment for each. The sample mean investment is $135,000 with astandard deviation of $30,000. Is there enough evidence to support your claim at α = 0.05? Use a P-value.


Solution: Hypothesis Testing Using P-values

• H0:

• Ha:

• α = • Test Statistic:

μ = $143,260

μ ≠ $143,2600.05

135,000 143,260

30,000 30

1.51

xz

n

µσ

−=

−=

= −

• Decision:

At the 5% level of significance, there is not sufficient evidence to conclude the mean franchise investment is different from $143,260.

P = 2(0.0655) = 0.1310

0.1310 > 0.05Fail to reject H0

z0-1.51

0.0655


• P-value = 2*NormCdf(-E99,-1.51)

Section 7.2

Z-Test for μUsing your Calculator

• Stat Test Z-test

If you have Data enter it in a list (in L1 for example)

Otherwise:

• Enter the Hypothesized population Mean.

• Enter sample mean, sample or pop. stand. Dev., and n.

• Choose the appropriate Alternative Hypothesis.

• And VOILA!!

Remember: If the P-value is Small, Reject the Null!!!

• Make your conclusion.44


Section 7.3

Hypothesis Testing for the Mean (Small Samples)



• Use the t-test to test a mean μ

• Use technology to find P-values

• Compare P-value to Alpha and make your Decision (Reject or Fail to Reject the Null Hypothesis)

• Make your Conclusion


t-Test for a Mean μ (n < 30, σ Unknown)

t-Test for a Mean

• A statistical test for a population mean.

• The t-test can be used when the population is normal or nearly normal, σ is unknown, and n < 30.

• The standardized test statistic is t.xts n

µ−=


Example: Using P-values with t-Tests

The American Automobile Association claims that the mean daily meal cost for a family of four traveling on vacation in Florida is $118. A random sample of 11 such families has a mean daily meal cost of $128 with a standard deviation of $20. Is there enough evidence to reject the claim at α = 0.10? Assume the population is normally distributed. (Adapted from American Automobile Association)


Solution: Using P-values with t-Tests

• H0:

• Ha:

• Decision:

μ = $118μ ≠ $118

TI-83/84set up: Calculate: Draw:

At the 0.10 level of significance, there is not enough evidence to reject the claim that the mean daily meal cost for a family of four traveling on vacation in Florida is $118.

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0.1664 > 0.10 Fail to reject H0.


Section 7.3

t-Test for μUsing your Calculator

Stat Test t-test

If you have Data enter it in a list (in L1 for example)

Otherwise:

• Enter the Hypothesized population Mean.

• Enter sample mean, sample stand. Dev., and n.


• And VOILA!!


• Make your conclusion.50Math 117 – Eddie Laanaoui

Section 7.4

Hypothesis Testing for Proportions



• Use z-test to test a population

proportion p


z-Test for a Population Proportion

z-Test for a Population Proportion

• A statistical test for a population proportion.

• Can be used when a binomial distribution is given such that np ≥ 5 and nq ≥ 5.

• The test statistic is the sample proportion .

• The standardized test statistic is z.

0p̂ pzpq n−=

p̂


Section 7.4

1 Prop Z-Test for pUsing your Calculator


• Stat Test 1 Prop Z-test

• Enter the Hypothesized population proportion.

• Enter x and Enter n.


• And VOILA!!

• Decision (reject or fail to reject the Null Hypothesis)


• Make your conclusion.

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